$f(x) = \dfrac{ 8 }{ \sqrt{ 5 - \lvert x \rvert } }$ What is the domain of the real-valued function $f(x)$ ?
Solution: First, we need to consider that $f(x)$ is undefined anywhere where the radicand (the expression under the radical) is less than zero. So we know that $5 - \lvert x \rvert \geq 0$ This means $\lvert x \rvert \leq 5$ , which means $-5 \leq x \leq 5$ Next, we need to consider that $f(x)$ is also undefined anywhere where the denominator is zero. So we know that $\sqrt{ 5 - \lvert x \rvert } \neq 0$ , so $\lvert x \rvert \neq 5$ This means that $x \neq 5$ and $x \neq -5$ So we have four restrictions: $x \geq -5$ $x \leq 5$ $x \neq -5$ , and $x \neq 5$ Combining these four, we know that $x > -5$ and $x < 5$ ; alternatively, that $-5 < x < 5$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid -5< x <5\, \}$.